44.3.23 problem 4(c)

Internal problem ID [9133]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.4 First Order Linear Equations. Page 15
Problem number : 4(c)
Date solved : Tuesday, September 30, 2025 at 06:05:46 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} x y^{\prime }+2&=x^{3} \left (y-1\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 22

ode:=x*diff(y(x),x)+2 = x^3*(y(x)-1)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\operatorname {LambertW}\left (c_1 \,{\mathrm e}^{\frac {1}{x^{2}}}\right ) x^{2}+1}{x^{2}} \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 85

ode=x*D[y[x],x]+2==x^3*(y[x]-1)*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {(-1)^{2/3} \left (y(x) x^2+2 x^2-1\right )}{\sqrt [3]{2} \left (y(x) x^2-x^2-1\right )}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]=\frac {(-2)^{2/3}}{9 x^2}+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.801 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*(y(x) - 1)*Derivative(y(x), x) + x*Derivative(y(x), x) + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - W\left (C_{1} e^{\frac {1}{x^{2}}}\right ) + \frac {1}{x^{2}} \]